de Broglie: matter waves
By 1924, physics faced an odd asymmetry. Einstein's photoelectric work (previous lesson) had forced physicists to accept that light — understood as a wave for a century, ever since Young's double-slit experiment demonstrated interference — also behaves like a stream of particles, each photon carrying momentum p = E/c = hf/c = h/λ. Louis de Broglie, then a doctoral student, asked the question nobody had: if waves can behave like particles, why shouldn't particles behave like waves? In his 1924 thesis he proposed extending the photon's momentum-wavelength relation to every particle of matter — electrons, atoms, baseballs — with no change to the formula at all.
De Broglie's hypothesis makes an unambiguous, testable prediction: an electron with momentum p should show wave behavior — interference, diffraction — with the same wavelength a photon of that momentum would have. In 1927, Clinton Davisson and Lester Germer, studying electron scattering off a nickel crystal at Bell Labs, found sharp, angle-dependent intensity peaks in the reflected electron beam — a diffraction pattern, not the smooth scattering a classical particle beam should produce. The angles matched exactly what de Broglie's λ = h/p predicted for the nickel lattice spacing, using the electrons' known momentum from the accelerating voltage. G. P. Thomson independently confirmed the same wave behavior with a transmission-diffraction setup the same year — earning de Broglie the 1929 Nobel Prize and Thomson (son of J. J. Thomson, who'd won his own Nobel for discovering the electron as a particle) a share of the 1937 prize for showing it was also a wave.
De Broglie's relation also retroactively explained something that had been a bare assumption in Bohr's 1913 model of the atom: why an orbiting electron is only allowed at certain radii. If an electron in orbit is really a wave, a stable orbit needs the wave to close smoothly on itself — an integer number of wavelengths fitting exactly around the circumference, 2πr = nλ. Substituting de Broglie's λ = h/p turns that into precisely Bohr's quantization condition, angular momentum in integer multiples of ħ, with no extra postulate needed. And the wavelength formula immediately explains why nobody had ever noticed matter waves before de Broglie: p = mv means λ shrinks as mass grows, and for anything larger than an atom the wavelength is spectacularly, unmeasurably small — you'll compute exactly how small for a thrown baseball in this unit's problem set. Electrons, with their tiny mass, are the sweet spot where matter-wave effects sit squarely in the reach of a tabletop experiment.